On Wilson loops for two touching circles with opposite orientation

Abstract

We study the Wilson loops for contours formed by a consecutive passage of two touching circles with a common tangent, but opposite orientation. The calculations are performed in lowest nontrivial order for ${\cal N}=4$ SYM at weak and strong coupling and for QCD at weak coupling. After subtracting the standard linear divergence proportional to the length, as well the recently analysed spike divergence, we get for the renormalised Wilson loops $\mbox{log}~W_{\mbox{\scriptsize ren}}=0$. The result holds for circles with different radii and arbitrary angle between the discs spanned by them.